Integrand size = 14, antiderivative size = 66 \[ \int x \cosh (a+b x) \coth (a+b x) \, dx=-\frac {2 x \text {arctanh}\left (e^{a+b x}\right )}{b}+\frac {x \cosh (a+b x)}{b}-\frac {\operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac {\operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2}-\frac {\sinh (a+b x)}{b^2} \]
-2*x*arctanh(exp(b*x+a))/b+x*cosh(b*x+a)/b-polylog(2,-exp(b*x+a))/b^2+poly log(2,exp(b*x+a))/b^2-sinh(b*x+a)/b^2
Time = 0.05 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.11 \[ \int x \cosh (a+b x) \coth (a+b x) \, dx=\frac {b x \cosh (a+b x)+b x \log \left (1-e^{a+b x}\right )-b x \log \left (1+e^{a+b x}\right )-\operatorname {PolyLog}\left (2,-e^{a+b x}\right )+\operatorname {PolyLog}\left (2,e^{a+b x}\right )-\sinh (a+b x)}{b^2} \]
(b*x*Cosh[a + b*x] + b*x*Log[1 - E^(a + b*x)] - b*x*Log[1 + E^(a + b*x)] - PolyLog[2, -E^(a + b*x)] + PolyLog[2, E^(a + b*x)] - Sinh[a + b*x])/b^2
Result contains complex when optimal does not.
Time = 0.44 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.33, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {5973, 3042, 26, 3777, 3042, 3117, 4670, 2715, 2838}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x \cosh (a+b x) \coth (a+b x) \, dx\) |
\(\Big \downarrow \) 5973 |
\(\displaystyle \int x \sinh (a+b x)dx+\int x \text {csch}(a+b x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -i x \sin (i a+i b x)dx+\int i x \csc (i a+i b x)dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int x \csc (i a+i b x)dx-i \int x \sin (i a+i b x)dx\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle i \int x \csc (i a+i b x)dx-i \left (\frac {i x \cosh (a+b x)}{b}-\frac {i \int \cosh (a+b x)dx}{b}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle i \int x \csc (i a+i b x)dx-i \left (\frac {i x \cosh (a+b x)}{b}-\frac {i \int \sin \left (i a+i b x+\frac {\pi }{2}\right )dx}{b}\right )\) |
\(\Big \downarrow \) 3117 |
\(\displaystyle i \int x \csc (i a+i b x)dx-i \left (\frac {i x \cosh (a+b x)}{b}-\frac {i \sinh (a+b x)}{b^2}\right )\) |
\(\Big \downarrow \) 4670 |
\(\displaystyle i \left (\frac {i \int \log \left (1-e^{a+b x}\right )dx}{b}-\frac {i \int \log \left (1+e^{a+b x}\right )dx}{b}+\frac {2 i x \text {arctanh}\left (e^{a+b x}\right )}{b}\right )-i \left (\frac {i x \cosh (a+b x)}{b}-\frac {i \sinh (a+b x)}{b^2}\right )\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle i \left (\frac {i \int e^{-a-b x} \log \left (1-e^{a+b x}\right )de^{a+b x}}{b^2}-\frac {i \int e^{-a-b x} \log \left (1+e^{a+b x}\right )de^{a+b x}}{b^2}+\frac {2 i x \text {arctanh}\left (e^{a+b x}\right )}{b}\right )-i \left (\frac {i x \cosh (a+b x)}{b}-\frac {i \sinh (a+b x)}{b^2}\right )\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle i \left (\frac {2 i x \text {arctanh}\left (e^{a+b x}\right )}{b}+\frac {i \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}-\frac {i \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2}\right )-i \left (\frac {i x \cosh (a+b x)}{b}-\frac {i \sinh (a+b x)}{b^2}\right )\) |
I*(((2*I)*x*ArcTanh[E^(a + b*x)])/b + (I*PolyLog[2, -E^(a + b*x)])/b^2 - ( I*PolyLog[2, E^(a + b*x)])/b^2) - I*((I*x*Cosh[a + b*x])/b - (I*Sinh[a + b *x])/b^2)
3.5.7.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x _Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x )], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[Cosh[(a_.) + (b_.)*(x_)]^(n_.)*Coth[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(c + d*x)^m*Cosh[a + b*x]^n*Coth[a + b* x]^(p - 2), x] + Int[(c + d*x)^m*Cosh[a + b*x]^(n - 2)*Coth[a + b*x]^p, x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(138\) vs. \(2(63)=126\).
Time = 0.63 (sec) , antiderivative size = 139, normalized size of antiderivative = 2.11
method | result | size |
risch | \(\frac {\left (b x -1\right ) {\mathrm e}^{b x +a}}{2 b^{2}}+\frac {\left (b x +1\right ) {\mathrm e}^{-b x -a}}{2 b^{2}}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{2}}+\frac {\operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {\ln \left ({\mathrm e}^{b x +a}+1\right ) x}{b}-\frac {\ln \left ({\mathrm e}^{b x +a}+1\right ) a}{b^{2}}-\frac {\operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {2 a \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b^{2}}\) | \(139\) |
1/2*(b*x-1)/b^2*exp(b*x+a)+1/2*(b*x+1)/b^2*exp(-b*x-a)+1/b*ln(1-exp(b*x+a) )*x+1/b^2*ln(1-exp(b*x+a))*a+polylog(2,exp(b*x+a))/b^2-1/b*ln(exp(b*x+a)+1 )*x-1/b^2*ln(exp(b*x+a)+1)*a-polylog(2,-exp(b*x+a))/b^2+2/b^2*a*arctanh(ex p(b*x+a))
Leaf count of result is larger than twice the leaf count of optimal. 255 vs. \(2 (61) = 122\).
Time = 0.25 (sec) , antiderivative size = 255, normalized size of antiderivative = 3.86 \[ \int x \cosh (a+b x) \coth (a+b x) \, dx=\frac {{\left (b x - 1\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (b x - 1\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (b x - 1\right )} \sinh \left (b x + a\right )^{2} + b x + 2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) - 2 \, {\left (b x \cosh \left (b x + a\right ) + b x \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - 2 \, {\left (a \cosh \left (b x + a\right ) + a \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 2 \, {\left ({\left (b x + a\right )} \cosh \left (b x + a\right ) + {\left (b x + a\right )} \sinh \left (b x + a\right )\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) + 1}{2 \, {\left (b^{2} \cosh \left (b x + a\right ) + b^{2} \sinh \left (b x + a\right )\right )}} \]
1/2*((b*x - 1)*cosh(b*x + a)^2 + 2*(b*x - 1)*cosh(b*x + a)*sinh(b*x + a) + (b*x - 1)*sinh(b*x + a)^2 + b*x + 2*(cosh(b*x + a) + sinh(b*x + a))*dilog (cosh(b*x + a) + sinh(b*x + a)) - 2*(cosh(b*x + a) + sinh(b*x + a))*dilog( -cosh(b*x + a) - sinh(b*x + a)) - 2*(b*x*cosh(b*x + a) + b*x*sinh(b*x + a) )*log(cosh(b*x + a) + sinh(b*x + a) + 1) - 2*(a*cosh(b*x + a) + a*sinh(b*x + a))*log(cosh(b*x + a) + sinh(b*x + a) - 1) + 2*((b*x + a)*cosh(b*x + a) + (b*x + a)*sinh(b*x + a))*log(-cosh(b*x + a) - sinh(b*x + a) + 1) + 1)/( b^2*cosh(b*x + a) + b^2*sinh(b*x + a))
\[ \int x \cosh (a+b x) \coth (a+b x) \, dx=\int x \cosh ^{2}{\left (a + b x \right )} \operatorname {csch}{\left (a + b x \right )}\, dx \]
Time = 0.26 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.42 \[ \int x \cosh (a+b x) \coth (a+b x) \, dx=\frac {{\left ({\left (b x e^{\left (2 \, a\right )} - e^{\left (2 \, a\right )}\right )} e^{\left (b x\right )} + {\left (b x + 1\right )} e^{\left (-b x\right )}\right )} e^{\left (-a\right )}}{2 \, b^{2}} - \frac {b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )}{b^{2}} + \frac {b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )}{b^{2}} \]
1/2*((b*x*e^(2*a) - e^(2*a))*e^(b*x) + (b*x + 1)*e^(-b*x))*e^(-a)/b^2 - (b *x*log(e^(b*x + a) + 1) + dilog(-e^(b*x + a)))/b^2 + (b*x*log(-e^(b*x + a) + 1) + dilog(e^(b*x + a)))/b^2
\[ \int x \cosh (a+b x) \coth (a+b x) \, dx=\int { x \cosh \left (b x + a\right )^{2} \operatorname {csch}\left (b x + a\right ) \,d x } \]
Timed out. \[ \int x \cosh (a+b x) \coth (a+b x) \, dx=\int \frac {x\,{\mathrm {cosh}\left (a+b\,x\right )}^2}{\mathrm {sinh}\left (a+b\,x\right )} \,d x \]